In a food processor, finely chop 1 cup of the shrimp. With a chef's knife, coarsely chop remaining shrimp.

In a bowl, beat eggs with a fork. Add shrimp, 3/4 cup of the breadcrumbs, salt, and reserved cooked vegetables. Stir well to combine. Cover and refrigerate 30 minutes.

In same nonstick skillet, cook andouille until lightly browned, 8 minutes. Drain on paper towels; let cool. In a small bowl, stir andouille and mayonnaise.

Heat remaining 1 cup of the oil in a skillet over medium heat until hot. Spread remaining 1 cup of the breadcrumbs on a piece of waxed paper. Shape shrimp mixture into 6 long oval patties, about 3/4-inch thick. Gently coat patties with breadcrumbs. Cook patties 5 minutes per side, or until browned and cooked through; drain on paper towels.

Spread andouille mayonnaise on cut sides of bread. Place lettuce, burgers, and tomatoes on bottom halves of bread. Replace tops.